Quadratic Differentials

1. Front Matter

   Pages I-XII

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2. Background Material on Riemann Surfaces

   • Kurt Strebel

   Pages 1-15

3. Quadratic Differentials

   • Kurt Strebel

   Pages 16-26

4. Local Behaviour of the Trajectories and the φ-Metric

   • Kurt Strebel

   Pages 27-37

5. Trajectory Structure in the Large

   • Kurt Strebel

   Pages 38-69

6. The Metric Associated with a Quadratic Differential

   • Kurt Strebel

   Pages 70-98

7. Quadratic Differentials with Closed Trajectories

   • Kurt Strebel

   Pages 99-150

8. Quadratic Differentials of General Type

   • Kurt Strebel

   Pages 151-178

9. Back Matter

   Pages 179-186

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A quadratic differential on aRiemann surface is locally represented by a ho­ lomorphic function element wh ich transforms like the square of a derivative under a conformal change of the parameter. More generally, one also allows for meromorphic function elements; however, in many considerations it is con­ venient to puncture the surface at the poles of the differential. One is then back at the holomorphic case. A quadratic differential defines, in a natural way, a field of line elements on the surface, with singularities at the critical points, i.e. the zeros and poles of the differential. The integral curves of this field are called the trajectories of the differential. A large part of this book is about the trajectory structure of quadratic differentials. There are of course local and global aspects to this structure. Be­ sides, there is the behaviour of an individual trajectory and the structure deter­ mined by entire subfamilies of trajectories. An Abelian or first order differential has an integral or primitive function is in general not single-valued. In the case of a quadratic on the surface, which differential, one first has to take the square root and then integrate. The local integrals are only determined up to their sign and arbitrary additive constants. However, it is this multivalued function which plays an important role in the theory; the trajectories are the images of the horizontals by single valued branches of its inverse.

• Derivative
• Extremale quasikonforme Abbildung
• Meromorphic function
• Quadratisches Differential
• Riemann surfaces
• Riemannsche Fläche

• Mathematisches Institut, Universität Zürich, Zürich, Switzerland

  Kurt Strebel

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